Advances in Risk Management

46 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENT

Therefore,

(λ 1 +λ 2 )(y 1 +y 2 )=λ 1 y 2 +λ 2 y 1

=λ 1 PPλ 2 +λ 2 PPλ 1 +λ 1 Pη+λ 2 Pη

=−(λ 1 −λ 2 )PP(λ 1 −λ 2 )

≤ 0

Sinceλ 1 ≥0,λ 1 ≥0,y 1 ≥0, andy 2 ≥0, it follows that:

(λ 1 −λ 2 )PP(λ 1 −λ 2 )= 0

which implies

P(λ 1 −λ 2 )= 0

Therefore, the optimal solution to equation (2.9) is

η∗=η+Pλ 1 =η+Pλ 2

which completes the proof.

REFERENCES

Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) “Coherent Measures of Risk”,
Mathematical Finance, 9(3): 203–28.
Basak, S. and Shapiro, A. (2001) “Value-at-Risk-Based Risk Management: Optimal Policies
and Asset Prices”,Review of Financial Studies, 14(2): 371–405.
Follmer, H. and Schied, A. (2002) “Convex Measures of Risk and Trading Constraints”,
Finance and Stochastics, (6)4: 429–47.
Jarrow, R. (2002) “Put Option Premiums and Coherent Risk Measures”,Mathematical
Finance, 12(2): 125–34.
Luenberger, D.G. (1990)Introduction to Linear and Nonlinear Programming,2nd edn
(Reading, MA: Addison-Wesley).
Merton, R.C. (1998) “Applications of Option-Pricing: Twenty-Five Years Later”,American
Economic Review, 88(3): 323–49.
Rockafellar, R.T. and Ziemba, W.T. (2000) “Modified Risk Measures andAcceptance Sets”,
Discussion Paper, University of British Columbia.